Semigroup Forum

, Volume 86, Issue 1, pp 192–201 | Cite as

On explicit representation and approximations of Dirichlet-to-Neumann semigroup



In his book (Functional Analysis, Wiley, New York, 2002), P. Lax constructs an explicit representation of the Dirichlet-to-Neumann semigroup, when the matrix of electrical conductivity is the identity matrix and the domain of the problem in question is the unit ball in ℝ n . We investigate some representations of Dirichlet-to-Neumann semigroup for a bounded domain. We show that such a nice explicit representation as in Lax book, is not possible for any domain except Euclidean balls. It is interesting that the treatment in dimension 2 is completely different than other dimensions. Finally, we present a natural and probably the simplest numerical scheme to calculate this semigroup in full generality by using Chernoff’s theorem.


Dirichlet-to-Neumann operator Lax’s Dirichlet-to-Neumann semigroup γ-harmonic lifting 



We wish to thank Professor Ralph deLaubenfels who was the instigator of this method, for his collaboration with the first author which ends up with this paper.


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran
  2. 2.Laboratoire de MathématiquesUniversité de PoitiersChassneuil du Poitou CedexFrance

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